Finite Group Databasebeta

← #10690772

Subgroups — #10690772

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#407C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#72127C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9117—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×M16—21no—
9C2×#953—21no—
10C22×#217—21no—
11C2×#930C411yescharacteristic, complemented
12C2×#930C411yescharacteristic, complemented
13C2×#930C2211yescharacteristic
14C3×#407—24no—
15C2×C3×C22⋊C8—21no—
16C2×C3×C22⋊C8C2212yes—
17#5213—21no—
18#5213—21no—
19#5213—21no—
20#5212C411yescharacteristic, complemented
21#5212C411yescharacteristic, complemented
22#5212C2211yescharacteristic
23C2×C3⋊#95—21no—
24C2×C3⋊#95C2212yes—
25#4470—24no—
26C2×#407D311yescharacteristic, complemented
27#2847—31no—
28C2×#327—32no—
29C22⋊#95—31no—
30#1294—32no—
31C25×C3D411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—42no—
35C23×C3×C4—42no—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67—21no—
44C22×#67—21no—
45C23×Dic3—22no—
46C22×D12C2×C411yescharacteristic
47C2×C3×M16—24no—
48C2×C3×M16—42no—
49C2×C3×M16—42no—
50C22×C3×C8—22no—
51#953—44no—
52C2×#217—24no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—42no—
57C2×#217—42no—
58#930—22no—
59#930—22no—
60#930—42no—
61#930—42no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic
66C2×C4⋊Dic3C2×C411yescharacteristic
67C2×C4⋊Dic3C2×C411yescharacteristic
68C3×C22⋊C8—24no—
69C3×C22⋊C8—22no—
70C3×C22⋊C8D414yescomplemented
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—22no—
75C3⋊#95—22no—
76C3⋊#95—42no—
77C24×C4C2×D311yescharacteristic
78C22×M16—21no—
79C2×#121—31no—
80C2×#121—31no—
81C2×#121—31no—
82C2×#120—61no—
83C22×C22⋊C4—61no—
84#407—24no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8C2×D312yes—
87#386—61no—
88#386—61no—
89#386—61no—
90#380—31no—
91#380—31no—
92#380—31no—
93C2×#95—32no—
94C2×#95—61no—
95#327—64no—
96C24×C3—22no—
97C24×C3—22no—
98C24×C3—21no—
99C24×C3—21no—
100C24×C3—22no—
101C24×C3—22no—
102C24×C3—41no—
103C24×C3—41no—
104C24×C3C22⋊C411yescharacteristic
105C24×C3C22⋊C411yescharacteristic
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—48no—
128C22×C3×C4—48no—
129C22×C3×C4C22⋊C411yescharacteristic
130C22×C3×C4C22⋊C411yescharacteristic
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D412yes—
134C22×C3×C4#4112yes—
135C2×#67—42no—
136C2×#67—42no—
137C2×#67—42no—
138C2×#67—42no—
139C2×#67—42no—
140C2×#67—42no—
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—41no—
144C2×#67—41no—
145C22×Dic3—24no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—42no—
150C22×Dic3—42no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C2×D12—22no—
156C2×D12—41no—
157C2×D12—41no—
158C3×M16—48no—
159C2×C3×C8—24no—
160C2×C3×C8—42no—
161C2×C3×C8—42no—
162#217—48no—
163#217—42no—
164#217—42no—
165#217—42no—
166(C2×D3)⋊C4—42no—
167(C2×D3)⋊C4—42no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170C4⋊Dic3—22no—
171C4⋊Dic3—41no—
172C4⋊Dic3—41no—
173C25D1211yescharacteristic
174C22×D4—31no—
175C22×D4—61no—
176C22×D4—61no—
177C22×D4—61no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—62no—
183C23×C4C4×D311yescharacteristic
184C23×C4C4×D311yescharacteristic
185C23×C4D1211yescharacteristic
186C23×C4#6711yescharacteristic
187C23×C4#6711yescharacteristic
188C23×C4C22×D311yescharacteristic
189C2×M16—24no—
190C2×M16—42no—
191C2×M16—42no—
192C22×C8—22no—
193#121—62no—
194#121—62no—
195#121—122no—
196#121—122no—
197#120—124no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C22⋊C4—64no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—122no—
209C2×C22⋊C4—122no—
210C22⋊C8—24no—
211C22⋊C8—22no—
212C22⋊C8#6714yescomplemented
213#95—61no—
214#95—61no—
215#95—61no—
216#95—62no—
217#95—62no—
218#95—122no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—44no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3#9912yes—
246C23×C3#10412yes—
247C23×C3C2×C22⋊C411yescharacteristic
248C23×C3#12112yes—
249C23×C3AGL(1,ℤ/8)11yescharacteristic
250C23×C3AGL(1,ℤ/8)12yes—
251C23×C3#13711yescharacteristic
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4#9912yes—
278C2×C3×C4C4×D412yes—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12311yescharacteristic
281#67—84no—
282#67—84no—
283#67—82no—
284#67—82no—
285C2×Dic3—44no—
286C2×Dic3—44no—
287C2×Dic3—44no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—82no—
298C2×Dic3—82no—
299C2×Dic3—82no—
300C2×Dic3—82no—
301D12—82no—
302C3×C8—44no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—41no—
310C24—41no—
311C24—61no—
312C24(C2×D3)⋊C411yescharacteristic
313C24(C2×D3)⋊C411yescharacteristic
314C24C2×D1211yescharacteristic
315C2×D4—62no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—122no—
319C2×D4—122no—
320C2×D4—122no—
321C2×D4—122no—
322C2×D4—122no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—121no—
326C2×D4—121no—
327C2×D4—122no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—64no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—122no—
354C22×C4—122no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4—122no—
358C22×C4—122no—
359C22×C4(C2×D3)⋊C411yescharacteristic
360C22×C4(C2×D3)⋊C411yescharacteristic
361C22×C4C2×C4×D311yescharacteristic
362C22×C4#23512yes—
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6712yes—
365M16—48no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—62no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C22⋊C4—122no—
373C22⋊C4—128no—
374C22⋊C4—122no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C22⋊C4—122no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—48no—
395C22×C3—48no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3#38611yescharacteristic
405C22×C3C2×#9911yescharacteristic
406C22×C3C2×#9911yescharacteristic
407C22×C3#41711yescharacteristic
408C22×C3#41811yescharacteristic
409C22×C3C2×#10411yescharacteristic
410C22×C3#42111yescharacteristic
411C22×C3#46812yes—
412C22×C3#46912yes—
413C2×D3—81no—
414C2×D3—81no—
415C2×D3—81no—
416C2×D3—81no—
417C2×D3—81no—
418C2×D3—81no—
419C2×D3—81no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—41no—
425Dic3—84no—
426Dic3—81no—
427Dic3—81no—
428Dic3—81no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—24no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23#80612yes—
463C23#82512yes—
464C23#90812yes—
465C23#90811yescharacteristic
466C23#90911yescharacteristic
467C23C2×(C2×D3)⋊C411yescharacteristic
468C23#93012yes—
469D4—242no—
470D4—244no—
471D4—244no—
472D4—242no—
473D4—242no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—124no—
493C2×C4—124no—
494C2×C4—124no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—242no—
505C2×C4—242no—
506C2×C4—242no—
507C2×C4—242no—
508C2×C4#80612yes—
509C2×C4C4×#6712yes—
510C2×C4#92911yescharacteristic
511C2×C4#95311yescharacteristic
512C8—44no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—42no—
516C2×C3—42no—
517C2×C3#150411yescharacteristic
518C2×C3#150711yescharacteristic
519C2×C3#151011yescharacteristic
520C2×C3#151311yescharacteristic
521C2×C3#187111yescharacteristic
522C2×C3#189511yescharacteristic
523C2×C3#189811yescharacteristic
524D3—161no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—48no—
540C22—48no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—241no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22C2×#80611yescharacteristic
557C22C2×#80611yescharacteristic
558C22#513111yescharacteristic
559C22#513311yescharacteristic
560C22#513412yes—
561C22#513512yes—
562C22C2×#82511yescharacteristic
563C22#513911yescharacteristic
564C22#521311yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—41no—
570C4—244no—
571C4—241no—
572C4—241no—
573C4—241no—
574C3#911711yescharacteristic, complemented, Sylow
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—481no—
580C2#6997611yescharacteristic
581C2#6997711yescharacteristic
582C2#6997811yescharacteristic
583C2#6997911yescharacteristic
584C2#7154311yescharacteristic
585C2#7198411yescharacteristic
586C2#7199011yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#407C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenterC2×#407D3
hypocenterC3#9117
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#407C2semidirect
3. #72127C2semidirect
4. C2×#4470C2semidirect
5. #67675C2semidirect
11. C2×#930C4semidirect
12. C2×#930C4semidirect
20. #5212C4semidirect
21. #5212C4semidirect
26. C2×#407D3semidirect
70. C3×C22⋊C8D4semidirect
212. C22⋊C8#67semidirect
574. C3#9117semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#4072
#840322
#721272
C2×#44702
#676752
#91173